Formula for a summation

mech-eng

Problem Statement
I wonder if there is a formula for this summation: $\sum_{n=1}^5= \frac 1n$
Relevant Equations
I know some formulas for summations but I don't know any formula for this case

$\sum_{k=1}^n=\frac{n({n+1})}2$
I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result.

$\sum_{k=1}^{n=5}=\frac {n!}{n!k}$

Thank you.

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Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result.

$\sum_{k=1}^{n=5}=\frac{n!}{n!k}$

Thank you.
Since you have a simple sum of 5 numbers, what is preventing you from just doing the addition? Admittedly, you need to find a common denominator, but that should not be too hard.

In general, there is no known "closed-form" formula for the so-called harmonic number, defined as
$$H(n) = \sum_{k=1}^n \frac 1 k$$
However, there are simple approximate formulas whose performance becomes better as $n$ becomes larger.

mech-eng

Since you have a simple sum of 5 numbers, what is preventing you from just doing the addition?
It was just an example. Sum might be 20 or 30 numbers. Yes with just 5 numbers it is very easy and the common denominator could be 5!. Is that called an $\textrm {harmonic sum}$?

Meanwhile would you also explain why my fraction line does not appear in my post the first post? What is wrong with my latex code?

Thanks

WWGD

Science Advisor
Gold Member
Maybe a way of double-checking if the formula is right is using the fact that it is known that the sum will never be an integer.

Mark44

Mentor
I wonder if there is a formula for this summation: $\sum_{n=1}^5= \frac 1n$
I know some formulas for summations but I don't know any formula for this case

$\sum_{k=1}^n=\frac{n({n+1})}2$
I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result.

$\sum_{k=1}^{n=5}=\frac {n!}{n!k}$
None of your equations makes any sense, since you aren't including that thing being summed.
It's as if you asked someone to evaluate this integral: $\int_1^5$.

Is the first summation supposed to be $\sum_{n=1}^5 \frac 1n$? If so, I don't know of any formula, but it's pretty easy to add the five fractions.

For your second equation, it looks like what you meant is $\sum_{k=1}^n k =\frac{n({n+1})}2$, the sum of the first k positive integers.

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